Updated: May 9, 2019
If you've ever had salad dressing, peanut butter, or milk, then you've had an emulsion. An emulsion is when two liquids that wouldn't normally mix, come together. The most common example of this is oil and water. The two don't mix because of the attractive forces between the molecules of each substance. Water molecules are held very tightly together, whereas oil molecules are not. The result is that oil is less dense than water, more viscous, and in some cases, solid.
In my previous posts about foams, we learned that by having a molecule that has both a water-loving end and a fat-loving (water-hating) end can help to make foams more stable. The same goes for emulsions. Proteins, soaps, and detergents are great examples of emulsifying agents.
These types of dual-ended molecules can also be found in nature. They are most commonly found in cell walls. Cell walls are made up of a lipid bilayer, which are made of sheet of molecules called phospholipids. They arrange so that all of the water-loving ends point in the same direction. If you were to grind up the membrane and add oil, the fat-loving ends would point into the oil droplet, and the water-loving end would point outward. This would prevent the droplet from combining with others, allowing the oil to dissolve in water. A common phospholipid is lecithin, which is found in egg yolks. When making mayonnaise, the proteins and lecithin in the egg yolk, and the phospholipids in the mustard, serve to stabilize the emulsion.
Another stabilizer in emulsions is vegetable gum. According to Culinary Reactions, gums are "large, starch-like that form thick colloidal suspensions on water". The gums found in mustard and garlic help to stabilize the emulsions they are added to.
Serious Eats' Two-Minute Mayo (although this recipe calls for an immersion blender, you can use a food processor or regular blender instead. I would not recommend a Nutri Bullet, or something similar, since you can slowly add the oil.)
Fine Cooking's How to Make (and Fix) Emulsions
Alton Brown's Homemade Roasted Honey Peanut Butter